Chapter 11
Simple C* -Algebras
This chapter contains a variety of results related to simple C* -algebras. Our
first goal is to analyze some deep structural work of Blackadar and Kirchberg
on simple nuclear QD C* -algebras. The generalized inductive limit theory
necessary for this result is interesting in its own right, but our target is
Corollary 11.3.10. Next, we discuss what Popa calls a "local quantization"
technique; this gives an internal characterization of quasidiagonality for a
large class of algebras and a simple proof of the fact that injective II 1 -
factors are AFD (Theorem 11.4.8). In the final section we present Connes's
celebrated uniqueness theorem for injective II1-factors.11.1. Generalized inductive limitsBlackadar and Kirchberg introduced generalized inductive limits in hopes of
broadening the scope of classification techniques. While we certainly follow
many .of their ideas here, we also draw inspiration from free probability
theory and take a norm-microstate point of view whenever convenient.
Definition 11.1.1. Let (Am, l.Pn,m) be a sequence of C*-algebras and linear,
adjoint-preserving^1 maps l.Pn,m: Am -t An, m < n, such that l.Ps,n o l.Pn,m =
<p 8 ,m, whenever m < n < s. We say that (Am, l.Pn,m) is a generalized inductive
system if
(1) SUPn>m 111.Pn,m(x)ll < oo for all m EN and XE Am and
(2) for each k, any a, b E Ak and c > 0 there is an M such that for all
M < m < n we have 111.Pn,m(l.Pm,k(a)<pm,k(b))-<pn,k(a)<pn,k(b)ll < c.(^1) This assumption is not necessary, but it can always be arranged, so there's no harm in
making it.
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